Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 417-424
417
HOLOMORPHIC EXTENSION OF GENERALIZATIONS
OF H’ FUNCTIONS
RICHARD D. CARMICHAEL
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico 88003, U.S.A.
(Received March 31, 1985)
In recent analysis we have defined and studied holomorphic functions in
ABSTRACT.
n
tubes in
which generalize the Hardy H
consider functions f(z), z
p
where C is the finite union of open convex cones
TO(C)
Rn
values in
C,
+ iO(C),
’
n + iC,
C
i,
j
,m,
and which satisfy
We prove a holomorphic extension theorem in
the norm growth of our new functions.
which f(z), z e T
In this paper we
C
functions in tubes.
x+iy, which are holomorphic in the tube T
is shown to be extendable to a function which is holomorphlc in
where O(C) is the convex hull of C, if the distributional boundary
TCJ
of f(z) from each connected component
of T
C
are equal.
KEY WORDS AND PHRASES. Generalization of
Functions in be Domains, Holomorphic
Cxtension, Fourier-Laplace Transform, Edge of the Wedge Theorem.
1980 AMS SUBJECT CLASSIFICATION CODE. 32A07, 32A10, 32A25, 32A35, 32A40, 46F20.
i.
INTRODUCTION.
The purpoue of this paper is to prove a holomorphic extension theorem (edge of
n and which
the wedge theorem) for functions which are holomorphic in a tube in
satisfy a norm growth condition that generalizes the norm growth for H p functions in
tubes.
The basis for the analysis presented here is the analysis in our papers
I-2 ].
Carmichael
We begin by stating some needed definitions.
(0,0
vertex at the origin
scalars
A.
n)
O) in
A regular cone ls an open convex cone C such that
entire straight llne.
0 for all y
C};
C*
The dual cone
C*
is always closed and convex (Vladlmirov
and is denoted pr(C).
T
n+
C for all positive
does not contain any
n
C*
{t
[3, p. 218]).
n:
<t,y> >
The
is called the projection of C
The function
Uc(t
is the indicatrix of
is a cone (with
Ay
of a cone C is defined as
intersection of the cone C with the unit sphere in
C
Rn
A set C
if y e C implies
.the
iC is a tube In
sup
ygpr(C) (-<t,y>)
cone C, and we note that
n.
C*
{t
t
Rn Uc(t)
_< 0}.
The set
The convex hull (convex envelope) of a cone C will be
denoted by O(C), and O(C) is also a cone.
OC
Put
C,
sup
Uo(c) (t)
tC,
Uc(t)
Rnc *
the number
418
R. D. CARHICHAEL
.
characterizes the nonconvextty of the cone C (Vladimirov [3, [..
Following
Vladtmtrov [4, p. 930] we say that a cone C :; 1(n with interior points has an admissible set of vectors if there are vectors e
C,
1, k
1,2
,n, which form
k
a basis for Rn; equivalently we say that such a set of n vectors in C is admissible
for the cone C.
Let B denote a proper open subset of
Let 0 < p <
and A > O. Let d(y)
lekl
Rn.
n.
B)
denote the distance from y c B to the complement of B in I
The space sP(T
(Carmtchael [1, pp. 80-81]), TB
1(n + tB, is the set of all functions f(z), z
x+ ty
T
which are holomorphtc in TB and whlch satisfy
B,
If(x+iy) lP
l[f(x+iy)[ILp
_<
M (l+
dx
_<
(I.1)
(d(y))-r) exp(2Alyl),
y
B,
_> 0 which can depend on f, p, and A but not on y B
M(f,p,A,r,s) which can depend on f, p, A, r, and s but not
for some constants r _> 0 and s
and for some constant M
sP(T
We defined and studied the functions
B) in Carmichael [1-2]. The
) were defined to generalize the H p functions in tubes (Stein and Weiss
[5, Chapter III]) and to contain the previous generalizations of the H p functions of
on y E B.
spaces
S(T
Vladimirov [6] and Carmichael and Hayashi [7].
We proved in Carmichael [I, Theorem 4.1, p. 92] that if B is a proper open
B) 1 < p <_ 2, A >_ O, has
then any element f(z) E
B
Fourier-Laplace integral representation for z E T in terms of a function g(t) which
connected subset of
n
sP(T
satisfies certain norm growth properties. In addition we proved in Carmlchael [i,
Corollary 4.1, p. 93] that if B
C, an open convex cone in Rn, then f(x+ly) has a
unique boundary value as
y/, y
E
C, in the strong topology of
’,
the space of
tempered distributions.
In this paper we prove a holomorphic extension theorem (edge of the wedge theore
for holomorphic functions in T C which satisfy (i.I) for y E C where C is a finite
union of open convex cones in
n;
the extended function is holomorphic in T O(C) where
To obtain our extension theor we use the information
O(C) is the convex hull of C.
from Carmlchael [1] which is contained in the preceding paragraph.
We proceed to the result of this paper after making the following definition;
the subspace
of
I <_ p <
is defined to be the set of all measurable
,
P
n,
functions g(t), t
-b
((1
such that there exists a real number b
0 for which
e p (Carmlchael [i, p. 83]).
g(t))
All subsequent notation and terminology in thls paper are that of Carmlchael [1-2].
+ ItlP)
HOLOMORPHIC EXTENSION.
Let C be an open cone in
2.
open convex cones
C
tubular cone T
y
g
Cj,
j
n
n and m
n+
m
is a positive integer.
I,
J
,m, are
Let f(z) be holomorphic in the
iC and satisfy (i.i) for y E C and for i < p
2.
For ay
m, the distance from y to the boundary of C is larger than or equal
I,
to the distance from y to the boundary of
1 < p <--2, j
_
n such that C jl cj where the Cj,
i
measurable functions
m.
gj(t)
Cj
from which it follows that f(z)
S(TCJ),-
Thus by Carmchael [I, Corollary 4.1, p. 93] there exist
E
q,
(l/p)+ (l/q)
i, with
supp(gj)
t:
Ucj(t)
_< A}
HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF H P FUNCTIONS
419
almost everywhere such that
n
f(z)
z e T
exp(2i<z,t>) dt,
gj(t)
cj
(2.1)
m,
i
j
pointwise and
’
Jim
y+
[gj]
e
[gj]
being the
f(x+iy)
yeC.
with
in the strong topology of
J
1
(2.2)
m,
Fourier transform of
We now state and prove the main result of this paper.
TttEOREM.
Rn
Let C be an open cone in
open convex cones, C
which is the union of a finite number of
.=
Cj
,.’
corresponding to each connected component
(O(C))*
such that
contains interior points and has an
x+iy, be holomorphtc in the tubular cone
admissible set of vectors. Let f(z), z
T C and satisfy (i.i) for y e C and i < p _< 2. Let the boundary values of f(x+iy) in
the strong topology of
of C
O(C)
m,
Then there is a function F(z) which is holomor-
given in (2.2) be equal in
phic in T
l,
Cj,
f(z), z e T C, where F(z) is of the form
P(z) H(z), z e T O(C)
and which satisfies V(z)
F(z)
S, (TO(C)),
(TO(C)
(l/p) +
P(z) being a polynomial in z and H(z) e $2A
0c
PC
I.
(l/q)
PROOF. By hypothesis the boundary values in (2.2) above are equal in
with
Since the Fourier transform is a topological isomorphism of
.
the elements
(l/p) + (l/q)
gj(t)
I
q
first paragraph of this section satisfy
in
We have
We call this common value g(t) and have g(t)
Uc(t)
Uo(c)(t),
tlon of 0 we have
c
j=l
C*,
t e
U0(c) (t) <_
0C
(Vladimirov [3, p. 220]) here we have
_< 0
C
From (2.3) and the fact that
we have that g e
let t
{t:
t e
Uc.(t),
m
eIn.
q’ (l/p) + (l/q)
i.
Now
(2.4)
Uo(c)(t)
max
j=l
supp(gj)
t e
Uc.(t)
m
{t:
< 0
C
C*.
Uc(t)
Ucj(t)
t e
n.
From (2.4) we now
1n
(2 5)
_< A} almost everywhere, j =l
m,
m
qC
Uo(c)(t)
(2.3)
(Vladimirov [3, p. 219, (54)]); and from the defniSince i ! 0 <
Uc(t), t e C, Rn
C
obtain
Uo(c) (t)
_
max
Uc(t)
m, obtained in the
gm(t)
g2(t)
gl(t)
we have that
onto
I
vanishes on
> A 0
C
};
A O <
C
jlo= {t: Ucj(t)> A}
as a distribution.
for such a point t we have by (2.5) that
Uo(c) (t)
S_ O
max
C
j=l
and hence
max
j=l
m
u
> A.
C (t)
m
Ucj (t)
Now
R. D. CARMICHAEL
420
Thus if t e {t:
Uo(c)(t) > A OC then
Since {t:
Uo(c)(t) <_ A0C
vanishes.
’
and {t:
N(;A
with
C)
0
< A O
C
Uo(c)(t)
{t:
j--i
> A} and on this latter set g
Ucj(t)
is a closed set in
{t:
supp(g)
in
t
U0(c)(t)
n we thus have
(2.6)
< A 0
C
(O(C))* + N(;A0C)
being the closure of the open ball in
(Vladimirov [4, Lemma i, p. 936])
n centered at
and with radius
(O(C))*
is closed and convex and by
Recall from section 1 that the dual cone
A 0
C
hypothesis in this Theorem (0(C))* contains interior points and has an admissible set
has order 0 then by Vladimlrov [4, Theorem i, p. 930]
of vectors. Since g
q=
n
g(t)
k=l
where
{ek}= I
n which
e
{t:
supp(G)
q
_
2
]G(t)
(2.7)
G(t)
is an admissible set of vectors for the cone
tinuous function of t
of g e
gradient>
<ek,
is unique corresponding to
(O(C))*,
n
{ek}k= 1
and the order 0
U0(c)(t) _< AOC}= (O(C))* + N(O;AO C)
Itl),
< K (i +
t
G(t) is a con-
and
n
(2.8)
n.
(In Vladlmlrov [4, Theorem i, p. 930]
((O(C))*) * O(C) (Vladlmlrov
the term
[3, p. 218]) should have non-empty interior (Vladlmlrov [4, p. 930]) which is certainly the case in this Theorem.) Since G(t) is continuous on n, then supp(G)
_< A0
{t:
as a function (Schwartz [8, Chapter i, sections 1 and 3]). (This
C
fact is also obtained in the proof of Vladlmlrov [4, Theorem I], and the containment
< A
{t:
supp(G)
PC which is stated preceding to (2.8) gives the support
such that for
t e
C
of G(t) as a function.) We now choose a function A(t)
is a constant
of nonnegatlve integers ID%(t) <_ M, t e n, where
any n-tuple
where the constant K is independent of t
"acute"
in our present situation means that
U0(c)(t)
Uo(c)(t)
_
n,
M
> 0, %(t)
1 for t on an
which depends only on ; and for
_< A
{t:
0 for t e
and %(t)
but not on a 2
{t:
Uo(c)(t)
u0(c)(t)
PC
_< A 0
C
as a function of t
F(z)
fn
n
(Carmlchael [i, p. 94]).
n for z e T0(C)
From (2.7) and supp(C,)
(3.1), p. 931])
{t:
Uo(c)(t)
F(z)
kl
where
H(z)
I
{t:U0(c)
We have that (%(t) exp(2i<z,t>))
Recalling (2.6) we now put
(n g(t) (t) exp(2l<z,t>)
g(t) exp(2i<z,t>) dt
neighborhood of
neighborhood of
<_ A O
c
dt,
z E T
O(C)
(2.9)
as a function we have (Vladlmirov [4,
<e k, -2iz> 2
H(z), z e
C(t) exp(2i<z,t>) dr, z
(2.10)
T O(C)
(2.11)
c}
(t)<_ 0
From the continuity of G(t) and (2.8) we easily have G(t)
for all p, i <_ p <
P
this combined with the support of G(t) as a function and Carmichael [I, Theorem 6.],
p. 98] yield
(exp(-2y,t>) G(t))
and
L
p,
y e O(C),
(2.12)
HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF H P FUNCTIONS
iexp(-2w<Y, t>)
(d(y))-r)
M (i +
s
exp(2A0C
IYl),
Y
421
(2.]3)
O(C)
.
s(G,p,A) _> 0, and M M(G,p,A,r,s) > 0, which are
r(G,p,A) _> 0, s
Then (2.12), (2.13), and
independent of y e 0(C), and for all p, 1 <_ p <
(T0(C)), (l/p) + (l/q) i,
Carmlchael [I, Theorem 5.1, p. 97] prove H(z) e S
for constants r
0C
for all p, 1 < p _< 2, and in particular H(z) E
defined in (2.9) is holomorphlc In T
O(C),
S2
A
0C
(TO(C)).
Then by (2.10), F(z)
and of course (2.10) is the desired repre-
sentation of F(z) in the statement of the Theorem where the polynomial
n
P(z)
k=l
and H(z) e S
2
A0C
S 0C (T0(C)),
(TO(C))
<ek,
-2Iz>
P(z) is
2
(l/p) + (l/q)
i, is given in (2.11).
By
(2.3), (2.6), and the definition of %(t) preceding (2.9), we see that (2.1) can be
rewritten as
n
f(z)
g(t) %(t) exp(2i<z,t>) dt
n
g(t) exp(2wi<z t>) dt
J
z E T
1
,m.
These identities and (2.9) show that F(z) is the desired holomorphlc extension of f(z)
to
The proof of the Theorem is complete.
T 0(C) and F(z)
f(z), z T
We emphasize that cones C exist for which the hypotheses of the Theorem are
C.
(0(C))*,
satisfied corresponding to C and
and examples are easily constructed.
If
0(C) in the Theorem is regular (i.e. if 0(C) does not contain an entire straight llne
in this case since O(C) is open and convex) then the interior of (O(C))* is not empty;
(O(C))* has an admissible set of vectors.
In the Theorem we have desired to obtain a result in which the holomorphlc
the Theorem applies in this case if
(T0(C))
extension function could be represented in terms of an S
happens under the assumptions on
(0(C))*
in the Theorem.
space; this
Under these assumptions we
were able to conclude that the continuous function G(t) in the representation (2.7)
had pointwlse support in {t:
_< A0 }. From this fact we were able to use
C
Carmlchael [i, Theorem 6.1] and then Carmlchael [I, Theorem 5.1] to obtain that H(z)
in (2.11) belongs to S
(TO(C)), (i/p) + (l/q) I, for all p, 1 < p < 2; and hence
_
Uo(c)(t)
the desired representation of the holomorphlc extension function F(z) was obtained in
(2.10).
From the proof of the Theorem the common value g(t) e
{t:
i < p < 2, in (2.3) has supp(g)
_< A0
in
supp(g)
Uo(c)(t)
C
’
I,
q, (l/p) + (l/q)
(recall (2.6)). If
is contained in this set almost everywhere as a function as well then the
restrictions on
(0(C))*
in the Theorem can be deleted in obtaining a holomorphic
extension result as we show in the following corollary.
COROLLARY i.
Let C be an open cone in
Rn
which is the union of a finite number
m
of open convex cones, C
C
j=l
J
Let f(z)
z
x+ iy be holomorphic
in the tubular
C
cone T and satisfy (i.i) for y e C and i < p < 2. Let the boundary values of f(x+iy)
in the strong topology of
corresponding to each connected component Cj
’
R. D. CARMICHAEL
422
_
and let this comon value g(t)
m, of C given in (2.2) be equal in
1
j
support in {t:
_< A0
C
Uo(c)(t)
’).
almost everywhere (as well as in
Then there i
F(z) which is holomorphic in TO(C) and which satisfies F(z)
2
(TO(C)).
S
2, F(z)
a function
and if p
A0C
hay,
f(z), z
E
TC;
Proceeding as in the proof of the Theorem we obtain the common value
< A0
in
I from (2.3) and supp(g)C {t:
(l/p) + (l/q)
g(t) g
C
{
t:
By our assumption supp(g)
<_ A 0C almost everywhere; thus by Carmichael
[I, Theorem 6.1, p. 98], g(t) satisfies
PROOF.
__’
’.
Uo(c)(t)
Uo(c)(t)
(exp(-2<y,t>) g(t)) c L
q,
y
(2.14)
O(C),
and
lexp(-2<Y, t>) g(t)
llLq
n
(d(y))-r) s
+
exp(2nA0C
s(g,q,A) >_ 0, and M
r(g,q,A) >_ O, s
for constants r
independent of y E O(C).
F(z)
< M (i
IYl)
O(C)
Y
(2.15)
M(g,q,A,r,s) > 0 which are
Then by Carmichael [I, Theorem 3.1, pp. 84-85] the function
g(t) X(t) exp(2i<z t>) dt
g(t) exp(2i<z t>) dt
(2.16)
z
T O(C) where %(t)
g C is the function defined in the proof of the
As in the proof of the Theorem F(z) is the desired holomorphic extension of
2 then q
2; in this case (2.14), (2.15), and Carmichael [I,
f(z) to T O(C)
If p
is holomorphic in
Theorem.
Theorem 5.1, p. 97] yield that F(z) E S
A
0C
(TO(C)).
The proof is complete.
We have a more general holomorphic extension theore
Here 0(C) is as general as possible and we make no assumption on the
Corollary i.
constructed g(t) in (2.3).
S
0c
than either the Theorem or
We lose the explicit information on F(z) being in an
(T 0(C)) space however.
COROLLARY 2.
Let the open cone C and the function f(z) be as in the hypothesis
of Corollary i with I < p
Let the boundary values of f(x+ ly) in the strong
2.
corresponding to each connected component
topology of
given in (2.2) be equal in
’.
such that F(z)
T
PROOF.
f(z), z
]
i,
m, of C
Then there is a holomorphic function F(z) in
TO(C)
C.
Define F(z), z E
TO(C),
i, is the common value in (2.3) in
as in (2.16) where g
and
C
supp(g)_ {t :Uo(c) (t) <_ AO C
(l/p) + (l/q)=
in
’
from
the proof of the Theorem. Then F(z) is holomorphic in T O(C) by the necessity of
Vladlmlrov [3, Theorem 2, p. 239] and is the desired holomorphlc extension of f(z) to
T O(C) because of (2.3) and (2.1). (Recall the proof of the Theorem.) The proof is
complete.
[3, Theorem 2, p. 239] that F(z) in Corollary 2 does
satisfy a polntwise growth estimate; but we cannot conclude that F(z) is in an
(T O(C) space for any p in Corollary 2.
Notice from Vladimlrov
S Oc
.
In the Theorem and Corollaries i and 2 the holomorphic extension function F(z),
z E T0(C), is defined by (2.9) (i.e. (2.16)) where g(t)
(l/p) + (l/q)= i,
{t:
<_ A 0
and supp(g)
in
Since O(C) is an open convex cone then in
Uo(c)(t)
C
:
P
HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF H FUNCTIONS
423
each of the results we can also conclude that
Rim
y-
[g]
F(x+iy)
(2.17)
E
y0(c)
in the strong topology of
’[g]
4.1, p. 93]" here
cone, A
’
by the boundary value proof in Carmtehael [1, Corollary
Fourier transform.
is the
Further, if 0(C) is a regular
2, in Coroilary 1 then we can conclude in Corollary 1 that
0, and p
<’[g],
F(z)
<’[g],
K(z-t)>
t), z a
Q(z
0(C),
(2.I8)
by CarchaeI [1, Corollarv 4.2, p. 91 here [g] is the boundary vaiue in
(2.17) and K(z- t) and Q(z; t) are the Cauchy and Poisson kerneis (eartchael [1,
(Recali fro the sentence
p. 83]), respectively, corresponding to the tube T O(c)
in
2
preceding the statement of Caichael [i, Corollary 4.2, p. 94] that
L2
[g]
If the cone C is (0,) or (-o, 0) or (-,0)
IYl,
d(y)
C, in (i.i).
Y
(-,0)
for C
implies
(
(0,).
(O,m)
in i dimension then of course
We have the following interesting result in 1 dimension
Note that
(0(C))*
{0} here which does
not have interior
points; so the following result is like Corollary 2.
Let f(z) be holomorphic in
COROLLARY 3.
(1.1) for 1 < p
R1 +
iC, C
(-m,O)
(O,m), and satisfy
Let the boundary values of f(x+iy) in the strong topology of
2.
’.
l+
from the upper and lower half planes given in (2.2) be equal in
entire holomorphic function F(z) such that
(-,).
First note that O(C)
< p ! 2, as in Corollary 2 and define
PROOF.
l,
F(z)
as in
exp(2i<z t>) dt
I (t)
(2.16).
Here
(O(C))*
i
F(z)
f(z), z
Obtain g(t)
iC.
(l/p) + (i/)
q
g(t) l(t) exp(2i<z t>) dt
{0} and supp(g)_ {t:
Uo(c)(t)
!
’
Then there is an
AOC}=
z e
(2.19)
(O(C))* +
has compact support here, and hence
[-A Oc,A OC]. Thus g
N(O;A 0C)
q
F(z) in (2.19) is the Fourier-Laplace transform of a distribution of compact support
(HSrmander [9, Theorem 1.7.5, p. 20]).
and hence is an entire holomorphlc function in
F(z)
+ iC, as before.
f(z), z
i
i
3.
ACKNOWLEDGEMENT.
The author expresses his sincere appreciation to the Department of Mathematical
Sciences of New Mexico State University for the opportunity of serving as Visiting
Professor during 1984-1985.
The author’s permanent address is Department of Mathematics and Computer Science,
Wake Forest University, Winston-Salem, North Carolina
27109, U.S.A.
This material is based upon work supported hy the National Science Foundation under
Grant No. DMS-8418435.
REFERENCES
I.
2.
p
CAP,MICHAEL, Richard D. Ceneralization of H Functions in Tubes.l,
Variables Theory Appl. 2 (1983), 79-101.
p
CARMIcHAEL, Richard D. Generalization of H Functions
Variables Theory Appl. 2 (1984), 243-259.
in Tubes. II,
Complex
Complex
R. D. CARMICHAEL
424
3.
4.
5.
6.
VLADIMIROV, V. S. Methods of the Theory of Functions of Many Complex Variables,
M.I.T. Press, Cambridge, Mass., 1966.
VLADIMIROV, V. S. Generalized Functions with Supports Bounded by the Side of An
Acute Convex Cone, Siberian Math. J. 9 (1968), 930-937.
STEIN, E. M. and G. WEISS. Introduction to Fourier Analysis on Euclidean Spaces,
Princeton University Press, Princeton, N.J., 1971.
VLADIMIROV, V. S. On Cauchy-Bochner Representations, Math. USSR-Izv. 6 (1972),
529-535.
7.
CARMICHAEL, Richard D. and Elmer K. HAYASHI. Analytic Functions in Tube Which
Are Representable by Fourier-Laplace Integrals, Pacific J. Math. 90 (1980),
51-61.
8.
SCHWARTZ, L. Thorie Des Distributions, Hermann, Paris, 1966.
9.
HMANDER,
1963.
L. Linear Partial
Different|al
Operators, Springer-Verlag, New York,